Finite Group Databasebeta

← #27280167

Subgroups — #27280167

Order 13568.

578 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20280459C211yesmaximal, characteristic
2#20277317C211yesmaximal, characteristic, complemented
3#20273283C211yesmaximal, characteristic
4C2×#16810348C211yesmaximal, characteristic, complemented
5#20272583C211yesmaximal, characteristic, complemented
6#20272582C211yesmaximal, characteristic, complemented
7C53×#1519C211yesmaximal, characteristic
8C2×#13133198—21no—
9C22×#11719919C2211yescharacteristic
10C2×C22⋊#88322C2211yescharacteristic
11#16810554C412yescomplemented
12#16810546—22no—
13#16810537C412yescomplemented
14#16810510—22no—
15#16810504—22no—
16#16810503—21no—
17#16810484—21no—
18#16810483C2211yescharacteristic
19#16810483C2211yescharacteristic
20C22⋊C4×Dic53—21no—
21#16810357—22no—
22#16810348—22no—
23C2×#121×C53C2211yescharacteristic
24C2×C22⋊C8×C53C2211yescharacteristic
25C53×#380C2211yescharacteristic
26C53×#327—22no—
27C23×Dic53D411yescharacteristic
28C23×Dic53D411yescharacteristic
29C22×#88322—21no—
30#13133198—42no—
31#13133198—42no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919—22no—
39C2×#11719919C2×C412yes—
40C2×#11719919D412yes—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53—22no—
44C2×C22⋊Dic53—22no—
45C2×C4×Dic53—22no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48#13133124—22no—
49#13133124—24no—
50#13133124—22no—
51#13133124C2×C412yes—
52#13133124D412yes—
53C22×D4×C53—21no—
54C22×D4×C53C2×C411yescharacteristic
55C23×C4×C53C2311yescharacteristic
56C22×C8×C53—21no—
57#121×C53—21no—
58#121×C53—21no—
59#121×C53—21no—
60#121×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53C2×C411yescharacteristic
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C22⋊C8×C53—21no—
66C22⋊C8×C53—21no—
67#95×C53D412yes—
68C24×C53—21no—
69C24×C53C2×D411yescharacteristic
70C2×D4×C53—21no—
71C2×D4×C53—21no—
72C2×D4×C53—41no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C2×D4×C53—41no—
76C2×D4×C53—41no—
77C2×D4×C53—41no—
78C2×D4×C53—81no—
79C22×C4×C53—21no—
80C22×C4×C53—22no—
81C22×C4×C53—81no—
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C22×C4×C53#4111yescharacteristic
88C22×Dic53—21no—
89C22×Dic53—21no—
90C22×Dic53—21no—
91C22×Dic53—21no—
92C22×Dic53—21no—
93C22×Dic53—21no—
94C22×Dic53—21no—
95C22×Dic53—21no—
96C22×Dic53—22no—
97C22×Dic53—22no—
98C22×Dic53—81no—
99C22×Dic53C22⋊C412yes—
100C22×Dic53C22⋊C412yes—
101C2×C8×C53—41no—
102C2×C8×C53—81no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—81no—
106C22⋊C4×C53—41no—
107C22⋊C4×C53—41no—
108C22⋊C4×C53—41no—
109C22⋊C4×C53—41no—
110C22⋊Dic53—44no—
111C22⋊Dic53—41no—
112C22⋊Dic53—41no—
113C22⋊Dic53—41no—
114C22⋊Dic53—41no—
115C22⋊Dic53—82no—
116C4⋊Dic53—81no—
117C4⋊Dic53—81no—
118#11719919—22no—
119#11719919—22no—
120#11719919—22no—
121#11719919—22no—
122#11719919—44no—
123C4×Dic53—82no—
124C2×#88322—41no—
125C2×#88322—81no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53—42no—
129C23×C53—42no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53C2×C22⋊C411yescharacteristic
134C23×C53#12011yescharacteristic
135C23×C53#12311yescharacteristic
136C23×C53#13511yescharacteristic
137C23×C53#13511yescharacteristic
138C23×C53AGL(1,ℤ/8)11yescharacteristic
139C23×C53AGL(1,ℤ/8)11yescharacteristic
140D4×C53—81no—
141D4×C53—81no—
142D4×C53—81no—
143D4×C53—81no—
144D4×C53—81no—
145D4×C53—81no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—42no—
158C2×C4×C53C4×D411yescharacteristic
159C2×C4×C53C4×D411yescharacteristic
160C2×C4×C53#12111yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×Dic53—22no—
163C2×Dic53—22no—
164C2×Dic53—22no—
165C2×Dic53—22no—
166C2×Dic53—42no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—42no—
172C2×Dic53—42no—
173C2×Dic53—42no—
174C2×Dic53—42no—
175C2×Dic53—42no—
176C2×Dic53—42no—
177C2×Dic53—42no—
178C2×Dic53—81no—
179C8×C53—81no—
180#88322—81no—
181#9206—531nomaximal, Sylow
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—41no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—82no—
193C22×C53—82no—
194C22×C53—81no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53#38611yescharacteristic
198C22×C53#41611yescharacteristic
199C22×C53#41811yescharacteristic
200C22×C53#44911yescharacteristic
201C22×C53#44911yescharacteristic
202C22×C53#48211yescharacteristic
203C22×C53#48211yescharacteristic
204C4×C53—21no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—82no—
209C4×C53—81no—
210Dic53—42no—
211Dic53—42no—
212Dic53—82no—
213Dic53—82no—
214#2618—531no—
215#2313—531no—
216#1519—531no—
217#1519D5311yescharacteristic
218C2×#324—531no—
219#1289—531no—
220#1289—531no—
221C2×C53—21no—
222C2×C53—21no—
223C2×C53—21no—
224C2×C53—21no—
225C2×C53—21no—
226C2×C53—21no—
227C2×C53—161no—
228C2×C53#187511yescharacteristic
229C2×C53#191211yescharacteristic
230C2×C53#191311yescharacteristic
231C2×#123—1061no—
232C2×#121C2×D5311yescharacteristic
233C22×C4⋊C4—531no—
234C2×C22⋊C8—531no—
235C2×C22⋊C8C2×D5311yescharacteristic
236#398—532no—
237#396—1062no—
238#394—532no—
239#387—1062no—
240#387—1062no—
241#386—1061no—
242#381—1061no—
243#380—531no—
244#380—531no—
245#380C2×D5311yescharacteristic
246C4×C22⋊C4—1061no—
247#327—22no—
248#327—1062no—
249#324—1062no—
250C53#920611yescharacteristic, complemented, Sylow
251C22×D4—21no—
252C22×D4C4×D5311yescharacteristic
253C23×C4—531no—
254C23×C4—531no—
255C23×C4C22×D5311yescharacteristic
256C22×C8—21no—
257C22×C8—1061no—
258#123—2122no—
259#123—2122no—
260#121—21no—
261#121—21no—
262#121—21no—
263#121—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—532no—
266C2×C4⋊C4—532no—
267C2×C4⋊C4—1061no—
268C2×C4⋊C4—1061no—
269C2×C4⋊C4—1061no—
270C2×C4⋊C4—1061no—
271C2×C4⋊C4—1061no—
272C2×C4⋊C4—1061no—
273C2×C4⋊C4—1062no—
274C2×C4⋊C4C4×D5311yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—1062no—
278C2×C22⋊C4—1062no—
279C2×C22⋊C4—1061no—
280C2×C22⋊C4—1061no—
281C2×C42—1062no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—1061no—
285C22⋊C8—1061no—
286#95—532no—
287#95—532no—
288#95—1062no—
289#95—1062no—
290#95—1064no—
291#95C22⋊D5312yes—
292C24—21no—
293C24D4×D5311yescharacteristic
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—81no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—81no—
306C22×C4—532no—
307C22×C4—532no—
308C22×C4—1061no—
309C22×C4—1061no—
310C22×C4—1061no—
311C22×C4—1061no—
312C22×C4—1061no—
313C22×C4—1061no—
314C22×C4—1061no—
315C22×C4—1061no—
316C22×C4—1062no—
317C22×C4—1062no—
318C22×C4—4241no—
319C22×C4C2×C4×D5311yescharacteristic
320C22×C4#1171994511yescharacteristic
321C22×C4D4×D5311yescharacteristic
322C22×C4D4×D5311yescharacteristic
323C22×C4#1171994711yescharacteristic
324C22×C4C2×C22⋊D5311yescharacteristic
325C2×C8—41no—
326C2×C8—81no—
327C2×C8—4241no—
328C2×C8—2121no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—81no—
332C4⋊C4—1062no—
333C4⋊C4—1062no—
334C4⋊C4—1062no—
335C4⋊C4—1062no—
336C4⋊C4—4241no—
337C4⋊C4—4241no—
338C4⋊C4—2124no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—4242no—
344C22⋊C4—2124no—
345C22⋊C4—2121no—
346C22⋊C4—2121no—
347C22⋊C4—2121no—
348C22⋊C4—2121no—
349C42—4242no—
350C23—41no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23C22⋊C4×D5311yescharacteristic
358C23#1313318511yescharacteristic
359C23#1313319911yescharacteristic
360C23#1313323611yescharacteristic
361C23#1313323811yescharacteristic
362C23#1313324111yescharacteristic
363C23#1313324411yescharacteristic
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—1062no—
383C2×C4—1062no—
384C2×C4—1062no—
385C2×C4—1062no—
386C2×C4—4241no—
387C2×C4—2122no—
388C2×C4—2122no—
389C2×C4—2122no—
390C2×C4—2122no—
391C2×C4—2122no—
392C2×C4—2122no—
393C2×C4—2122no—
394C2×C4—2121no—
395C2×C4—2121no—
396C2×C4—2121no—
397C2×C4—2121no—
398C2×C4—2122no—
399C2×C4#1313317711yescharacteristic
400C2×C4C4×C22⋊D5311yescharacteristic
401C2×C4#1313318811yescharacteristic
402C2×C4#1313319011yescharacteristic
403C8—81no—
404C8—4241no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—82no—
416C22—82no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22#1681049911yescharacteristic
421C22#1681061611yescharacteristic
422C22#1681062611yescharacteristic
423C22#1681076511yescharacteristic
424C22#1681076711yescharacteristic
425C22#1681094511yescharacteristic
426C22#1681094611yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—82no—
432C4—81no—
433C4—4242no—
434C4—4242no—
435C4—2122no—
436C4—2122no—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—161no—
444C2#2027503411yescharacteristic
445C2#2027524911yescharacteristic
446C2#2027525411yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1519C2
socleC22×C53#386
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenter#1519D53
hypocenterC53#9206
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20277317C2semidirect
4. C2×#16810348C2semidirect
5. #20272583C2semidirect
6. #20272582C2semidirect
11. #16810554C4semidirect
13. #16810537C4semidirect
250. C53#9206semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202804592
#202773172
#202732832
C2×#168103482
#202725832
#202725822
C53×#15192
#920653